| L(s) = 1 | + 92·3-s + 274·5-s + 7.20e3·7-s + 643·9-s + 4.53e4·11-s − 1.11e4·13-s + 2.52e4·15-s − 5.58e4·17-s − 4.88e5·19-s + 6.62e5·21-s + 2.53e5·23-s − 4.82e6·25-s + 2.58e6·27-s − 7.65e5·29-s + 8.18e6·31-s + 4.17e6·33-s + 1.97e6·35-s + 7.20e6·37-s − 1.02e6·39-s − 5.78e7·41-s + 3.73e7·43-s + 1.76e5·45-s + 2.92e7·47-s + 3.45e7·49-s − 5.13e6·51-s + 4.51e7·53-s + 1.24e7·55-s + ⋯ |
| L(s) = 1 | + 0.655·3-s + 0.196·5-s + 1.13·7-s + 0.0326·9-s + 0.934·11-s − 0.108·13-s + 0.128·15-s − 0.162·17-s − 0.860·19-s + 0.743·21-s + 0.189·23-s − 2.47·25-s + 0.934·27-s − 0.200·29-s + 1.59·31-s + 0.612·33-s + 0.222·35-s + 0.632·37-s − 0.0710·39-s − 3.19·41-s + 1.66·43-s + 0.00640·45-s + 0.873·47-s + 6/7·49-s − 0.106·51-s + 0.785·53-s + 0.183·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(6.692725189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.692725189\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{4} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 92 T + 869 p^{2} T^{2} - 360232 p^{2} T^{3} + 869 p^{11} T^{4} - 92 p^{18} T^{5} + p^{27} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 274 T + 980707 p T^{2} - 49324116 p^{2} T^{3} + 980707 p^{10} T^{4} - 274 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4124 p T + 7357311745 T^{2} - 210073691312248 T^{3} + 7357311745 p^{9} T^{4} - 4124 p^{19} T^{5} + p^{27} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 11158 T + 19430158919 T^{2} + 704160367295644 T^{3} + 19430158919 p^{9} T^{4} + 11158 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 55866 T + 176300156175 T^{2} + 36848383911863436 T^{3} + 176300156175 p^{9} T^{4} + 55866 p^{18} T^{5} + p^{27} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 488772 T + 894752133597 T^{2} + 278541391854157688 T^{3} + 894752133597 p^{9} T^{4} + 488772 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 253888 T + 3760899055749 T^{2} - 79946519271339136 T^{3} + 3760899055749 p^{9} T^{4} - 253888 p^{18} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 765318 T + 23825127102099 T^{2} + 51064343430823464708 T^{3} + 23825127102099 p^{9} T^{4} + 765318 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 8189680 T + 70383446448941 T^{2} - \)\(39\!\cdots\!20\)\( T^{3} + 70383446448941 p^{9} T^{4} - 8189680 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 7208194 T + 231033887217035 T^{2} - \)\(22\!\cdots\!52\)\( T^{3} + 231033887217035 p^{9} T^{4} - 7208194 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 57891986 T + 1501631602198007 T^{2} + \)\(27\!\cdots\!24\)\( T^{3} + 1501631602198007 p^{9} T^{4} + 57891986 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 37304708 T + 1468933201726561 T^{2} - \)\(32\!\cdots\!72\)\( T^{3} + 1468933201726561 p^{9} T^{4} - 37304708 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 29210992 T + 2427654375829853 T^{2} - \)\(41\!\cdots\!12\)\( T^{3} + 2427654375829853 p^{9} T^{4} - 29210992 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 45140082 T + 6203828668958763 T^{2} - \)\(12\!\cdots\!28\)\( T^{3} + 6203828668958763 p^{9} T^{4} - 45140082 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 205334804 T + 37468210900247829 T^{2} - \)\(36\!\cdots\!68\)\( T^{3} + 37468210900247829 p^{9} T^{4} - 205334804 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 109264870 T + 6601050775320759 T^{2} + \)\(81\!\cdots\!88\)\( T^{3} + 6601050775320759 p^{9} T^{4} + 109264870 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 52322348 T + 44506260843441049 T^{2} - \)\(20\!\cdots\!20\)\( T^{3} + 44506260843441049 p^{9} T^{4} - 52322348 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 150048136 T + 94610927281357973 T^{2} - \)\(82\!\cdots\!96\)\( T^{3} + 94610927281357973 p^{9} T^{4} - 150048136 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 761429998 T + 351523184728667207 T^{2} - \)\(10\!\cdots\!08\)\( T^{3} + 351523184728667207 p^{9} T^{4} - 761429998 p^{18} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 214389152 T + 268463825206964141 T^{2} - \)\(35\!\cdots\!32\)\( T^{3} + 268463825206964141 p^{9} T^{4} - 214389152 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 36110892 T + 499106543499529917 T^{2} - \)\(16\!\cdots\!68\)\( T^{3} + 499106543499529917 p^{9} T^{4} - 36110892 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 1319848398 T + 1218754159695802167 T^{2} - \)\(73\!\cdots\!60\)\( T^{3} + 1218754159695802167 p^{9} T^{4} - 1319848398 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 101766310 T + 569086063469878143 T^{2} + \)\(76\!\cdots\!88\)\( T^{3} + 569086063469878143 p^{9} T^{4} - 101766310 p^{18} T^{5} + p^{27} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47047773880724399742806108509, −9.987259165958110391921025164252, −9.612949499326054580371246046453, −9.532155544314191269285464681157, −8.757460387040241585910560510804, −8.583431477653097268943578254412, −8.387781956990450102513784531839, −7.967967812775452902675192303488, −7.52016050390039401278457434380, −7.21491536886443840724333174120, −6.61204259157735127532660092050, −6.27066226335895458818142787853, −6.05389477705569544985676927827, −5.29635640260511093028170745920, −5.01888592064525839522287960696, −4.57575606318265201946682052308, −4.10302815943744425449080250958, −3.58793691354702789587428192995, −3.51519825731533888436614331548, −2.42509905768944101526940301331, −2.23431673957311734383739711429, −2.05748788775294444255825316222, −1.25181985631209534396202508353, −0.924724256680914798463288417518, −0.39562879739202963381656370925,
0.39562879739202963381656370925, 0.924724256680914798463288417518, 1.25181985631209534396202508353, 2.05748788775294444255825316222, 2.23431673957311734383739711429, 2.42509905768944101526940301331, 3.51519825731533888436614331548, 3.58793691354702789587428192995, 4.10302815943744425449080250958, 4.57575606318265201946682052308, 5.01888592064525839522287960696, 5.29635640260511093028170745920, 6.05389477705569544985676927827, 6.27066226335895458818142787853, 6.61204259157735127532660092050, 7.21491536886443840724333174120, 7.52016050390039401278457434380, 7.967967812775452902675192303488, 8.387781956990450102513784531839, 8.583431477653097268943578254412, 8.757460387040241585910560510804, 9.532155544314191269285464681157, 9.612949499326054580371246046453, 9.987259165958110391921025164252, 10.47047773880724399742806108509
